In this course, one continues the study of derivatives and integrals begun in Calculus I & Calculus II. However, the perspective is broadened in several ways:

1. Real-valued functions of several variables are considered. Thus, a quantity z might depend on two variables z = f(x,y), or even more variables. This is necessary for the most realistic applications of calculus, as most things in real life do not depend on a single input. For two simple examples, in Physics, the pressure of a gas depends on both its temperature and its volume, written P = f(V,T) (exactly which function is used here depends on the setting: the ideal gas law expresses one possibility, while other models such as the van der Waals law might be more accurate). In economics, the Gross National Product, GNP, depends on literally thousands of independent variables, including the supplies of various raw materials, the unemployment rate, the demand of the consumers, prices of various commodities, etc. etc. There is probably no way to write down an explicit function in complete detail, the point we make is simply that it is some function of many variables GNP = f(w,x,y,....).

2. A function may have outputs which are vector quantities instead of numerical outputs. These are called vector-valued functions as opposed to real valued functions. There is a further breakdown:
   • A vector-valued function of a single real variable is also known as a parametric curve. For example in Calculus I one studies motion on a straight line (such as a car driving on a straight highway), where the position on the highway might be given as a function of time t: x = f(t). Now in multi-variable calculus, we generalize to motion in a plane, where both the x- & y-coordinates are functions of time: P(t) = (x(t), y(t)); i.e., the output is the vector P = (x,y), depending on the time t. Or the motion of an electron in a cloud chamber can take place in 3-space instead of a plane, so it's position is a vector with three coordinates: P(t) = (x(t), y(t), z(t)), each depending on the real number t.

   • A vector-valued function of several real variables is often used when both the input and the output have several coordinates. This is useful when both the magnitude and direction of the output depend on several quantities. For example, in fluid dynamics, one may represent the flow of a liquid or gas, such as wind, as a function. Here, the direction and the magnitude of the wind varies from point to point, so is an example of a 3-dimensional output which depends on a 3-dimensional input. Obviously such things come up in applications like meteorology, or in medicine, where one needs to know the details of blood flow in a human heart in order to design, say, a safe artificial valve.

3. Other coordinate systems are used freely, such as polar coordinates for the Euclidean plane, and cylindrical or spherical coordinates for Euclidean 3-space.

This broadening of the point of view has several practical consequences, paramount among them is the fact that the graphs of such functions no longer fit in a plane - more dimensions are needed. For example a function z = f(x,y) has a graph which is a surface in 3-space, if there are more independent variables, even more dimensions are required, making it difficult or impossible to visualize. Similarly, vector-valued functions usually require more dimensions, so an alternate way of visualizing them, known as "vector-fields" (particularly useful in Physics - electromagnetic fields are the classic example) are sometimes employed.

Another wrinkle is that in calculus I when computing limits, such as the limit of f(x) as x approaches a, one need only consider x approaching a from the left or from the right. When approaching a point in higher dimensions however, one can approach from infinitely many different directions; even worse, one can approach a point along a straight line or along some curved path, even one which is quite convoluted, spiralling around the target point for example! This makes it more difficult for limits to exist, and special care must be given to any definition which depends on the notion of a limit. This means we need to take a careful look at what it means for a function of several variables to be continuous, to be differentiable, to be integrable. We will discover some new twists in higher dimensions which do not occur in the more restricted situation of Calculus I & II.

In the end however, all the usual notions of Calculus will extend to the multivariable case, but usually with more variety in the possibilities. For example, in place of the derivative of a function, one has the partial derivatives (note the plural!), the directional derivative, and the "total derivative (also known as the gradient). On the integration side, we have iterated integrals, double interals (or triple integrals, etc.), potential functions (something like an antiderivative), line integrals, and surface integrals. All of these concepts have their uses (some of which will look quite familiar - for example there is a way to use the gradient of a function to find the maximum and minimum values of a function, and a "second derivative test" to distinguish between them, just as in Calculus I. But the real beauty in this material is in the new power it gives the calculus to handle applications unavailable in Calculus I or II, and also in the inter-relationships between these new concepts. In fact our goal will be to get to Gauss's theorem, which explores the relationship between a surface integral and a triple integral.

We will achieve these goals by covering Chapters 12 - 16 of our text. At times, it may be useful to look at supplementary material, such as on polar coordinates. Other times it would be useful to see some applications not covered in the text, such as the historical derivation of Kepler's laws of planetary motion from Newton's law of gravity, or more details on vector fields and electro-magnetism. Time permitting, when this material is covered, supplementary notes wil be provided, so that students need purchase only one text.

One last comment on the prerequisites: because of the presence of higher dimensions, because sometimes we consider vector-valued functions, it is seen as an advantage to know some linear algebra (hence MA200 is a pre-requisite). Indeed, for surfaces tangent planes replace the Calculus I notion of tangent lines, the gradient of a surface is a vector function, and the second derivative of a surface (or more generally, the total derivative of a vector function) is best viewed as a matrix. Even the chain rule is best understood as an example of matrix multiplication. This is not to say that Calculus III cannot be learned by someone ignorant of linear algebra, it is just that everything makes so much more sense when viewed from this perspective. (By the same token, after learning Calculus III, one often gains a better understanding of some of the ideas in linear algebra. The two complement each other.)

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